Signatures with Flexible Public Key: A Unified Approach to Privacy-Preserving Signatures (Full Version)

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1 Signatures with Flexible Public Key: A Unified Approach to Privacy-Preserving Signatures (Full Version) Michael Backes 1,3, Lucjan Hanzlik 2,3, Kamil Kluczniak 4, and Jonas Schneider 2,3 1 CISPA Helmholtz Center i.g., backes@cispa.saarland 2 CISPA, Saarland University, {hanzlik, jonas.schneider}@cispa.saarland 3 Saarland Informatics Campus, 4 The Hong Kong Polytechnic University, Department of Computing, kkklucz@polyu.edu.hk Abstract. We introduce a new cryptographic primitive called signatures with flexible public key. We divide the key space into equivalence classes induced by a relation R. A signer can efficiently change his key pair to a different representative of the same class, but without a trapdoor it is hard to distinguish if two public keys are related. This primitive offers a unified approach to the modular construction of signature schemes with privacy-preserving components. Namely, we show how to build the first ring signature scheme in the plain model without trusted setup, where signature size depends only sub-linearly on the number of ring members. Moreover, we show how to combine our primitive with structure-preserving signatures on equivalence classes (SPS-EQ) to construct static group signatures and self-blindable certificates. When properly instantiated, the result is a group signature scheme that has a shorter signature size than the current state-of-the-art scheme by Libert, Peters, and Yung from Crypto 15. In its own right, our primitive has stand-alone applications in the cryptocurrency domain. In particular it enables the straightforward implementation of so-called stealth addresses. Keywords: flexible public key, equivalence classes, stealth addresses, ring signatures, group signatures 1 Introduction Digital signatures aim to achieve two security goals: Integrity of the signed message and authenticity of the signature. A great number of proposals relax these goals or introduce new ones to accommodate the requirements of specialized application scenarios. As one example, consider sanitizable signatures [1] where the goal of preserving the integrity of the message is relaxed to allow for authorized

2 modification and redaction of the signed message. This paper introduces a novel characterization of authenticity. The goal is not a complete relaxation, such that any impostor can sign messages on behalf of a legitimate signer, but rather that authenticity holds with respect to some established legitimate signer, but who it is exactly remains hidden. Achieving the latter without enabling the former is one of the main challenges we tackle in this paper. Our new primitive, which we call signatures with flexible public key (SFPK) formalizes a signature scheme, where verification and signing keys live in a system of equivalence classes induced by a relation R. Given a signing or verification key it is possible to transform the key into a different representative of the same equivalence class, i.e., the pair of old key and new key is contained in relation R. Thus, we extend the requirement of unforgeability of signatures to the whole equivalence class of the given key under attack. However, an additional requirement we make is that it should be infeasible, without a trapdoor, to even check whether two keys are in the same class. This property, which we call computational class-hiding, ensures that given an old verification key, a signature under a fresh representative is indistinguishable from a signature under a different newly generated key, which lives in a different class altogether with overwhelming probability. Intuitively this means that signers can produce signatures for their whole class of keys, but they cannot sign for a different class (because of unforgeability) and they are able to hide which class the signature belongs to, i.e., to hide their own identity in the signature (because of class-hiding). The property of class-hiding is especially useful in cases where there is a (possibly pre-defined) set of known verification keys and a verifier only needs to know that the originator of a given signature was part of that set. Indeed, upon reading the first description of the scheme s properties, what should come to mind immediately is the setting of group signatures [13] and to some extent ring signatures [29] where the group is chosen at signing time and considered a part of the signature. Our primitive yields highly efficient, cleanly constructed group and ring signature schemes, but it should be noted, that SFPK on its own is neither of the two. The basic idea to build a group signature scheme from signatures with flexible public key is to combine them with an equally re-randomizable certificate on the signing key. Such a certificate is easily created through structure-preserving signatures on equivalence classes [23] by the group manager on the members verification key. A group signature is then produced by signing the message under a fresh representative of the flexible public key and tying that signature to the group by also providing a blinded certificate corresponding to the fresh flexible key. This fresh certificate can be generated from the one provided by the group manager. Opening of group signatures is done using the trapdoor that can be used to distinguish if public keys belong to the same equivalence class. In the case of ring signatures, the certification of keys becomes slightly more complex, since we cannot make any assumption on the presence of a trusted group manager. Therefore, the membership certificate is realized through a perfectly sound proof 2

3 of membership. The basic principle, however, remains the same, pointing to an elegant, unified approach to both group and ring signatures. Our contributions. This paper develops a new cryptographic building block from the ground up, presenting security definitions, concrete instantiations and applications. The main contributions are as follows: Signatures with flexible public key. Our new primitive is a natural abstraction and formalization of design principles that are already at the heart of many ring and group signature constructions found in the literature. Thus it offers a unified perspective on these two primitives and aids in modular design of efficient constructions by making explicit properties which have to be achieved by the identity-hiding component as outlined above. Generic constructions & tailored instantiations. We demonstrate how SFPK can be used to build group and ring signatures in a modularized fashion. For each construction, we give an efficient standard model SFPK instantiation which takes into account the differences in setting between group and ring signature. The resulting group and ring signature schemes have smaller (asymptotic and concrete) signature sizes than the previous state of the art schemes, including schemes with non-standard assumptions as long as one requires the strongest level of security. For instance, the static group signature scheme due to Libert, Peters, and Yung achieves fully anonymous signatures secure under standard non-interactive assumptions at a size of 8448 bits per signature. Our scheme based on comparable assumptions achieves the same security using 7680 bits per signature. Another variant of our scheme under an interactive assumption achieves signature sizes of only 3072 bits per signature, thus more than halving the size achieved in [25] and not exceeding by more than factor 3 the size of signatures in the scheme due to Bichsel et al. [6] which produces signatures of size 1280 bits but only offers a weaker form of anonymity under an interactive assumption in the random oracle model. A comprehensive comparison between our scheme and known group signature constructions can be found in Section 5.4. Our ring signature construction is the first to achieve signature sizes in O( N) without trusted setup and security under standard assumptions in the strongest security model by Bender, Katz and Morselli [5]. Thereby, we settle an issue that was stated as an open problem in the ASIACRYPT 2017 presentation of [27]. Applications of independent interest. Constructions of signatures with flexible public key that allow for a straightforward key recovery property lend themselves to numerous stand-alone applications in the field of cryptocurrencies. We exemplify this by showing how to implement stealth addresses for Bitcoin [30, 28], which allow a party to transfer currency to an anonymous address that the sender has generated from the receivers long-term public key. No interaction with the receiver is necessary for this transaction and the receiver can recover and subsequently spend the funds without linking them to their long-term identity. 3

4 1.1 Further Related Work At first glance, signatures with flexible public keys are syntactically reminiscent of structure-preserving signatures on equivalence classes[23]. While both primitives are similar in spirit, the former considers equivalence classes of key pairs while the latter only considers equivalence classes on messages. Another related primitive are signatures with re-randomizable keys[16]. The crucial difference to our new primitive is that re-randomization is akin to and indeed indistinguishable from sampling a fresh key from the whole key space. This means a signature scheme with re-randomizable keys cannot achieve class hiding and unforgeability under flexible public keys simultaneously. The ring signature built from signatures with flexible public keys is the first ring signature scheme to achieve signature size O( N) where N is the size of the ring without any trusted setup. With a trusted setup, constant size constructions are known, the most recent one being [27] which is based on signatures with rerandomizable keys and SNARKs. 2 Preliminaries We denote by y A(x, ω) the execution of algorithm A outputting y, on input x with randomness ω, writing just y $ A(x) if the specific randomness used is not important. We will sometimes omit the usage of random coin in the description of algorithms if it is obvious from the context (e.g. sampling group elements). The superscript O in A O means that algorithm A has access to oracle O. Moreover, we say that A is probabilistic polynomial-time (PPT) if A uses internal random coins and the computation for any input x {0, 1} terminates in polynomial time. By r $ S we mean that r is chosen uniformly at random over the set S. We will use 1 G to denote the identity element in group G, [n] to denote the set {1,..., n}, u to denote a vector and ( x 0... x x to denote the binary )bin representation of x. Definition 1 (Bilinear map). Let us consider cyclic groups G 1, G 2, G T of prime order p. Let g 1, g 2 be generators of respectively G 1 and G 2. We call e : G 1 G 2 G T a bilinear map (pairing) if it is efficiently computable and the following holds: Bilinearity: (S, T ) G 1 G 2, a, b Z p, we have e(s a, T b ) = e(s, T ) a b, Non-degeneracy: e(g 1, g 2 ) 1 is a generator of group G T, Definition 2 (Bilinear-group generator). A bilinear-group generator is a deterministic polynomial-time algorithm BGGen that on input a security parameter λ returns a bilinear group BG = (p, G 1, G 2, G T, e, g 1, g 2 ) such that G 1 = g 1, G 2 = g 2 and G T are groups of order p and e : G 1 G 2 G T is a bilinear map. Bilinear map groups with an efficient bilinear-group generator are known to be instantiable with ordinary elliptic curves introduced by Barreto and Naehrig [3] (in short BN-curves). 4

5 Invertible Sampling. We use a technique due to Damgård and Nielsen [15]: A standard sampler returns a group element X on input coins ω. A trapdoor sampler returns coins ω on input a group element X. Invertible sampling requires that (X, ω) and (X, ω ) are indistinguishably distributed. This technique was also used by Bender, Katz and Morselli [5] to prove full anonymity (where the adversary receives the random coins used by honest users to generate their keys) of their ring signature scheme. 2.1 Number Theoretical Assumptions In this section we recall assumptions relevant to our schemes. They are stated relative to bilinear group parameters BG := (p, G 1, G 2, G T, e, g 1, g 2 ) $ BGGen(λ). Definition 3 (Decisional Diffie-Hellman Assumption in G i ). Given BG and elements gi a, gb i, gz i G i it is hard for all PPT adversaries A to decide whether z = a b mod p or z $ Z p. We will use Adv ddh A (λ) to denote the advantage of the adversary in solving this problem. Definition 4 (Square Decisional Diffie-Hellman Assumption in G i [2]). Given BG and elements gi a, gz i G i it is hard for all PPT adversaries A to decide whether z = a 2 mod p or z $ Z p. We will use Adv sddh A (λ) to denote the advantage of the adversary in solving this problem. We now state the bilateral variant of the well known decisional linear assumption, where the problem instance is given in both G 1 and G 2. This definition was also used by Ghadafi, Smart and Warinschi [20]. Definition 5 (Symmetric Decisional Linear Assumption). Given BG, elements f 1 = g f 1, h 1 = g1 h, f1 a, h b 1, g1 z G 1 and elements f 2 = g f 2, h 2 = g2 h, f2 a, h b 2, g2 z G 2 for uniformly random f, h, a, b Z p it is hard for all PPT adversaries A to decide whether z = a + b mod p or z $ Z p. We will use Adv linear A (λ) to denote the advantage of the adversary in solving this problem. In this paper we use a variant of the 1-Flexible Diffie-Hellman assumption [26]. We show that this new assumption, which we call the co-flexible Diffie- Hellman (co-flex) assumption, holds if the decisional linear assumption holds. We also introduce a similar assumption called square-flexible Diffie-Hellman (sq-flex). Definition 6 (co-flexible Diffie-Hellman Assumption). Given BG, elements g1, a g1, b g1, c g1 d G 1 and g2, a g2, b g2, c g2 d G 2 for uniformly random a, b, c, d Z p, it is hard for all PPT adversaries A to output (g1) c r, (g1) d r, g1 r a b. We will use (λ) to denote the advantage of the adversary in solving this problem. Adv co-flexdh A Lemma 1. The co-flexible Diffie-Hellman assumption holds for BG if the decisional linear assumption holds for BG. 5

6 Definition 7 (sq-flexible Diffie-Hellman Assumption). Given BG, elements g1, a g1, b g1, c g1 d G 1 and g2, a g2, b g2, c g2 d G 2 it is hard for all PPT adversaries A to output (g1) c r, (g1) d r, g1 r2 a b. We will use Adv sq-flexdh A (λ) to denote the advantage of the adversary in solving this problem. Unfortunately, it is unknown whether for this assumption we can state a lemma similar to 1. However, under the Knowledge-of-Exponent (KEA) assumption [14], the sq-flexdh and co-flexdh assumptions are equivalent. This implies that the sq-flexdh holds in the generic group model. 2.2 Non-Interactive Proof Systems In this paper we make use of non-interactive proof systems. Although we define the proof system for arbitrarily languages, in our schemes we use the efficient Groth-Sahai (GS) proof system for pairing product equations [22]. Let R be an efficiently computable binary relation, where for (x, w) R we call x a statement and w a witness. Moreover, we will denote by L R the language consisting of statements in R, i.e. L R = {x w : (x, w) R}. Definition 8 (Non-Interactive Proof System). A non-interactive proof system Π consists of the following three algorithms (Setup, Prove, Verify): Setup(λ): on input security parameter λ, this algorithm outputs a common reference string ρ. Prove(ρ, x, w): on input common reference string ρ, statement x and witness w, this algorithm outputs a proof π. Verify(ρ, x, π): on input common reference string ρ, statement x and proof π, this algorithm outputs either accept(1) or reject(0). Some proof systems do not need a common reference string. In such a case, we omit the first argument to Prove and Verify. Definition 9 (Soundness). A proof system Π is called sound, if for all PPT algorithms A the following probability, denoted by Adv sound Π,A (λ), is negligible in the security parameter λ: Pr[ρ Setup(λ); (x, π) A(ρ) : Verify(ρ, x, π) = accept x L R ]. We say that the proof system is perfectly sound if Adv sound Π,A (λ) = 0. Definition 10 (Witness Indistinguishability (WI)). A proof system Π is witness indistinguishable, if for all PPT algorithms A we have that the advantage Adv wi Π,A(λ) computed as: Pr[ρ Setup(λ); (x, w 0, w 1 ) A(λ, ρ); π Prove(ρ, x, w 0 ) : A(π) = 1] Pr[ρ Setup(λ); (x, w 0, w 1 ) A(λ, ρ); π Prove(ρ, x, w 1 ) : A(π) = 1], where (x, w 0 ), (x, w 1 ) R, is at most negligible in λ. We say that the proof system if perfectly witness indistinguishable if Adv wi Π,A(λ) = 0. 6

7 Prove(x, w) 1 : ρ 1 := (f 1, f 2, h 1, h 2,...) $ Setup PPE (λ); r, s $ Z p 2 : ρ 2 := (f 1, f 2, h 1, h 2, f r 1, f r 2, hs 1, hs 2, gr+s 1, g r+s 2 ) 3 : π Linear $ Prove Linear((ρ 1, ρ 2), (r, s)) 4 : π 1 $ Prove PPE (ρ 1, x, w); π 2 $ Prove PPE (ρ 2, x, w) 5 : return π := (ρ 1, ρ 2, π Linear, π 1, π 2) Verify(x, π) 1 : parse π = (ρ 1, ρ 2, π Linear, π 1, π 2) 2 : return Verify PPE (ρ 1, x, π 1) = 1 3 : Verify PPE (ρ 2, x, π 2) = 1 4 : Verify Linear ((ρ 1, ρ 2), π Linear) = 1 Scheme 1: Perfectly Sound Proof System for Pairing Product Equations Perfectly Sound Proof System for Pairing Product Equations We briefly recall the framework of pairing product equations that is used for the languages of the Groth-Sahai proof system [22]. For constants A i G 1, B i G 2, t T G T, γ ij Z p which are either publicly known or part of the statement, and witnesses X i G 1, Y i G 2 given as commitments, we can prove that: n m m n e(a i, Y i ) e(x i, B i ) e(x i, Y i ) γij = t T. i=1 i=1 j=1 i=1 The system (Setup PPE, Prove PPE, Verify PPE ) has several instantiations based on different assumptions. In this paper we only consider the instantiation based on the symmetric linear assumption given by Ghadafi, Smart and Warinschi [20]. For soundness it must be ensured, that Setup PPE outputs a valid DLIN tuple. This can be enforced by requiring a trusted party performs the setup. However, in our schemes we require a proof system which is perfectly sound, even if a malicious prover executes the Setup PPE algorithm. To achieve this we use the ideas by Groth, Ostrovsky and Sahai [21]. The authors propose a perfectly sound and perfectly witness indistinguishable proof system (Prove Linear, Verify Linear ) which does not require a trusted setup. Using it one can show that given tuples T 1, T 2 as a statement, at least one of T 1 and T 2 is a DLIN tuple. The results were proposed for type 1 pairing but the proof itself is only given as elements in G 2. Moreover, our variant of the DLIN assumption gives the elements in both groups. Thus, we can apply the same steps as in [21]. The cost of such a proof is 6 elements in G 2. Next is the observation that the tuples T 1 and T 2 can each be used as common reference strings for the pairing product equation proof system. Since at least one of the tuples is a valid DLIN tuple, at least one of the resulting proofs will be perfectly sound. Witness-indistinguishability will be only computational, since we have to provide T 1 and T 2 to the verifier but that is sufficient in our case. The full scheme is presented in Scheme 1. The size of the proofs produced this way is 2 (3 e + 3 w 1 + 5) elements in G 1 and 2 (3 e + 3 w 2 + 5) + 6 elements in G 2, where e is the number of equations proven, w 1 is the number of witnesses in G 1 and w 2 is the number of witnesses in G 2. 7

8 Theorem 1. Scheme 1 is a perfectly sound proof system for pairing product equations if the system (Setup PPE, Prove PPE, Verify PPE ) is perfectly sound in the common reference string model. Proof (Sketch). Because Π Linear is perfectly sound Verify Linear ((ρ 1, ρ 2 ), π Linear ) = 1 means that at least one of ρ 1 and ρ 2 is a DLIN tuple. It follows that at least one of π 1 and π 2 is a perfectly sound proof for the statement x. Thus, statement x must be true. Theorem 2. Scheme 1 is a computational witness indistinguishable proof system if the system (Setup PPE, Prove PPE, Verify PPE ) is perfectly witness indistinguishable in the common reference string model. Proof (Sketch). Because the proof system for the pairing product equations is witness indistinguishable, we change the witness we use in proof π 1. Note that this change may include the change of ρ 1 to a non-dlin tuple but the proof π Linear is still valid because ρ 2 is a DLIN tuple. Next we replace ρ 1 with ρ 2 and use Setup PPE to compute ρ 2. Finally, we change the witness used to compute π Structure-Preserving Signatures on Equivalence Classes Hanser and Slamanig introduced a cryptographic primitive called structure-preserving signatures on equivalence classes [23]. Their work was further extended by Fuchsbauer, Hanser and Slamanig in [18] and [19]. The idea is simple but provides a powerful functionality. The signing Sign SPS (M, sk SPS ) algorithm defines an equivalence relation R that induces a partition on the message space. By signing one representative of a partition, the signer in fact provides a signature for all elements in it. Moreover, there exists a procedure ChgRep SPS (M, σ SPS, r, pk SPS ) that can be used to change the signature to a different representative without knowledge of the secret key. Existing instantiations allow to sign messages from the space (G i )l, for l > 1, and for the following relation R exp : given two messages M = (M 1,..., M l ) and M = (M 1,..., M l ), we say that M and M are from the same equivalence class (denoted by [M] R ) if there exists a scalar r Z p, such that i [l] (M i ) r = M i. Security Definition. We formally define structure-preserving signatures on equivalence classes as follows: Definition 11 (Structure-preserving signatures for equivalence relation R). A SPS-EQ scheme on (G i )l (for i {1, 2}) consists of the following algorithms: BGGen SPS (λ): a deterministic algorithm that on input a security parameters λ, outputs bilinear-group parameters BG. KGen SPS (BG, l): on input a parameter BG and a vector length l > 1, this probabilistic algorithm outputs a key pair (sk SPS, pk SPS ). 8

9 Sign SPS (M, sk SPS ): on input a message M (G i )l and secret key sk SPS, this probabilistic algorithm outputs a signature σ SPS on the equivalence class [M] R. ChgRep SPS (M, σ SPS, r, pk SPS ): on input a representative M of an equivalence class [M] R, signature σ SPS for M, scalar r and a public key pk SPS, this probabilistic algorithm returns an updated message-signature pair (M, σ SPS ), where M = (M) r (component-wise exponentiation) is the new representative and σ SPS its updated signature. Verify SPS (M, σ SPS, pk SPS ): on input a representative M, signature σ SPS and a public key pk SPS, this deterministic algorithm outputs 1 if σ SPS is a valid signature for M under public key pk SPS and 0 otherwise. VKey SPS (sk SPS, pk SPS ): on input a secret key sk SPS and public key pk SPS, this deterministic algorithm outputs 1 if both keys are consistent and 0 otherwise. The original paper defines two properties of SPS-EQ namely unforgeability under chosen-message attacks and class-hiding. Fuchsbauer and Gay [17] recently introduced a weaker version of unforgeability called unforgeability under chosen-open-message attacks, which restricts the adversaries signing queries to messages where it knows all exponents. Definition 12 (Signing Oracles). A signing oracle is an O SPS (sk SPS, ) (resp. O op (sk SPS, )) oracle, which accepts messages (M 1,..., M l ) (G i )l (resp. vectors (e 1,..., e l ) (Z p) l ) and returns signature under sk SPS on those messages (resp. on messages (g e1 1,..., ge l 1 ) (G i )l ). Definition 13 (EUF-CMA (resp. EUF-CoMA)). A SPS-EQ scheme (BGGen SPS, KGen SPS, Sign SPS, ChgRep SPS, Verify SPS, VKey SPS ) on (G i )l is called existentially unforgeable under chosen message attacks (resp. adaptive chosenopen-message attacks), if for all PPT algorithms A having access to an open signing oracle O SPS (sk SPS, ) (resp. O op (sk SPS, )) the following adversary s advantage (with templates T 1, T 2 defined below) is negligible in the security parameter λ: [ ] Adv l,t1 SPS-EQ,A (λ) = Pr BG BGGen SPS (λ); (sk SPS,pk SPS ) $ KGen SPS (BG,l); (M,σ SPS ) $ A O T 2 (sk SPS, ) (pksps ) : M Q. [M ] R [M] R Verify SPS (M,σ SPS,pk SPS )=1 where Q is the set of messages signed by the signing oracle O T2 and for T 1 = euf-cma we have T 2 = SPS, and for T 1 = euf-coma we have T 2 = op. A stronger notion of class hiding, called perfect adaptation of signatures, was proposed by Fuchsbauer et al. in [19]. Informally, this definition states that signatures received by changing the representative of the class and new signatures for the representative are identically distributed. In our schemes we will only use this stronger notion. Definition 14 (Perfect Adaption of Signatures). A SPS-EQ scheme on (G i )l perfectly adapts signatures if for all (sk SPS, pk SPS, M, σ, r), where, 9

10 VKey SPS (sk SPS, pk SPS ) = 1, M (G 1) l, r Z p and Verify SPS (M, σ, pk SPS ) = 1, the distribution of are identical. ((M) r, Sign SPS (M r, sk SPS )) and ChgRep SPS (M, σ, r, pk SPS ) 3 Signatures with Flexible Public Key We begin by introducing the idea behind our primitive. In the notion of existential unforgeability of digital signatures, the adversary must return a signature valid under the public key given to him by the challenger. Imagine now that we allow a more flexible forgery. The adversary can return a signature that is valid under a public key that is in some relation R to the public key chosen by the challenger. Similar to the message space of SPS-EQ signatures, this relation induces a system of equivalence classes on the set of possible public keys. A given public key, along with the corresponding secret key can be transformed to a different representative in the same class using an efficient, randomized algorithm. The adversary has access to this functionality by providing random coins which the challenger uses to change the representative before signing. Since there might be other ways of obtaining a new representative, the forgery on the challenge equivalence class is valid as long as the relation holds, even without knowledge of the explicit randomness that leads to the given transformation. Note, that the challenger thus needs a way to ascertain whether the forgery is valid, which cannot be verification through the transformation algorithm. Indeed, for the full definition of our schemes security we will require that it should not be feasible, in absence of the concrete transformation randomness, to determine whether a given public key belongs to one class or another. This property called class-hiding in the style of a similar property for SPS-EQ signatures should hold even for an adversary who has access to the randomness used to create the key pairs in question. The apparent conflict is resolved by introducing a trapdoor key generation algorithm TKeyGen which outputs a key pair (sk, pk) and a class trapdoor τ for the class the key pair is in. The trapdoor allows the challenger to reveal whether a given key is in the same class as pk, even if doing so efficiently is otherwise assumed difficult. Since we require that the keys generated using the trapdoor key generation and the regular key generation are distributed identically, unforgeability results with respect to one also hold with respect to the other. Definition 15 (Signature with Flexible Public Key). A signature scheme with flexible public key (SFPK) is a tuple of PPT algorithms (KeyGen, TKeyGen, Sign, ChkRep, ChgPK, ChgSK, Verify) such that: KeyGen(λ, ω): takes as input a security parameter λ, random coins ω coin and outputs a pair (sk, pk) of secret and public keys, 10

11 TKeyGen(λ, ω): a trapdoor key generation that takes as input a security parameter λ, random coins ω coin and outputs a pair (sk, pk) of secret and public keys, and a trapdoor τ. Sign(sk, m): takes as input a message m {0, 1} and a signing key sk, and outputs a signature σ, ChkRep(τ, pk): takes as input a trapdoor τ for some equivalence class [pk ] R and public key pk, the algorithm outputs 1 if pk [pk ] R and 0 otherwise, ChgPK(pk, r): on input a representative public key pk of an equivalence class [pk] R and random coins r, this algorithm returns a different representative pk, where pk [pk] R. ChgSK(sk, r): on input a secret key sk and random coins r, this algorithm returns an updated secret key sk. Verify(pk, m, σ): takes as input a message m, signature σ, public verification key pk and outputs 1 if the signature is valid and 0 otherwise. A signature scheme with flexible public key is correct if for all λ N, all random coins ω, r coin the following conditions hold: 1. The distribution of key pairs produced by KeyGen and TKeyGen is identical. 2. For all key pairs (sk, pk) $ KeyGen(λ, ω) and all messages m we have Verify(pk, m, Sign(sk, m)) = 1 and Verify(pk, m, Sign(sk, m)) = 1, where ChgPK(pk, r) = pk and ChgSK(sk, r) = sk. 3. For all (sk, pk, τ) $ TKeyGen(λ, ω) and all pk we have ChkRep(τ, pk ) = 1 if and only if pk [pk] R. Definition 16 (Class-hiding). For scheme SFPK with relation R and adversary A we define the following experiment: C-H A SFPK,R(λ) ω 0, ω 1 $ coin (sk i, pk i ) $ KeyGen(λ, ω i) for i {0, 1} m $ A(ω 0, ω 1) b $ {0, 1}; r $ coin sk $ ChgSK(sk b, r); pk $ ChgPK(pk b, r) σ $ Sign(sk, m) ˆb $ A(ω 0, ω 1, m, σ, pk ) return b = ˆb A SFPK is class-hiding if for all PPT adversaries A, its advantage in the above experiment is negligible: ] Adv c-h A,SFPK(λ) = [C-H Pr A SFPK,R(λ) = = negl(λ). Definition 17 (Existential Unforgeability under Flexible Public Key). For scheme SFPK with relation R and adversary A we define the following experiment: 11

12 EUF-CMA A SFPK,R(λ) ω $ coin (sk, pk, τ) $ TKeyGen(λ, ω); Q := (pk, m, σ ) $ A O1 (sk, ),O 2 (sk,, ) (pk, τ) return (m, ) Q ChkRep(τ, pk ) = 1 Verify(pk, m, σ ) = 1 O 1 (sk, m) σ $ Sign(sk, m) Q := Q {(m, σ)} return σ O 2 (sk, m, r) sk $ ChgSK(sk, r) σ $ Sign(sk, m) Q := Q {(m, σ)} return σ A SFPK is existentially unforgeable with flexible public key under chosen message attacks if for all PPT adversaries A the advantage in the above experiment is negligible: ] Adv euf cma A,SFPK [EUF (λ) = Pr CMA A SFPK(λ) = 1 = negl(λ). Definition 18 (Strong Existential Unforgeability under Flexible Public Key). A SFPK is strong existentially unforgeable with flexible public key under chosen message attacks if for all PPT adversaries A the advantage Adv seuf cma A,SFPK (λ) in the above experiment, where we replace the line (m, ) Q with (m, σ ) Q, is negligible. Finally, we define an optional property of SFPK signature schemes called key recovery. In a standard application, the public key and secret key are randomized by the signer. Obviously, the ChgPK algorithm can be executed by any third party using random coins r, which can be later shared with the signer. This way the signer can compute the corresponding secret key. For some application we would like to work without any interaction. It is easy to see that allowing the user to extract the new secret key only using his old secret key would break class-hiding. Fortunately, we can use the additional trapdoor returned by the TKeyGen algorithm. More formally, we define this optional property as follows. Definition 19 (Key Recovery Property). A SFPK has recoverable signing keys if there exists an efficient algorithm Recover such that for all security parameters λ N, random coins ω, r and all (sk, pk, τ) $ TKeyGen(λ, ω) and pk $ ChgPK(pk, r) we have ChgSK(sk, r) = Recover(sk, τ, pk ). 3.1 Flexible Public Key in the Multi-user Setting In this subsection, we address applications where a part of the public key of the user is generated by some trusted third party and is common among several users, e.g. the definition of the hash function used in Waters signatures. We therefore define an additional algorithm CRSGen that, given a security parameter, outputs a common reference string ρ. We assume that this string is an implicit input to 12

13 all algorithms. If the KeyGen is independent from ρ, we say that such a scheme supports key generation without setup. We will now discuss the implication of this new algorithm on the security definitions. Usually, we require that the common reference string is generated by an honest and trusted party (i.e. by the challenger in definitions 16 and 17). We additionally define those notions under maliciously generated ρ. We call a scheme class-hiding under malicious reference string if the class-hiding definition holds even if in definition 16 the adversary is allowed to generate the string ρ. Similarly, we call a SFPK scheme unforgeable under malicious reference string if the unforgeability definition 17 holds if ρ is generated by the adversary. 3.2 On Signatures with Re-Randomizable Keys Fleischhacker et al. [16] introduced signatures with re-randomizable keys, which allow a re-randomization of signing and verification keys such that re-randomized keys share the same distribution as freshly generated keys and a signature signed under a randomized key can be verified using an analogously randomized verification key. They also define a notion of unforgeability under re-randomized keys, which allows an adversary to learn signatures under the adversaries choice of randomization of the signing key under attack. The goal of the adversary is to output a forge under the original key or under one of its randomizations. Regular existential unforgeability for signature schemes is a special case of this notion, where the attacker does not make use of the re-randomization oracle. The difference to signatures with flexible public keys is that re-randomization in [16] is akin to sampling a fresh key from the space of all public keys, while changing the representative in our case is restricted to the particular key s equivalence class. Note that one might intuitively think that signatures under rerandomizable keys are just signatures with flexible keys where there is only one class of keys and because re-randomizing is indistinguishable from fresh sampling. In this case class hiding would be perfect. However, such a scheme cannot achieve unforgeability under flexible keys, since it would be enough for an attacker to sample a fresh keypair and use a signature under that key as the forgery. Another way of mapping signatures with re-randomizable keys to the flexible public key world would be to make the set of equivalence classes the singleton sets of all public keys, i.e. each key is the unique representative of its own equivalence class. This collapses unforgeability under flexible public keys to the standard unforgeability notion of digital signatures. In this case, however, class hiding would be impossible to achieve, since there is just one unique representative for each class. Note, that in the class-hiding definition, the challenge key pairs are not required to be in the same or separate classes. Therefore, even if both keys are from different classes, the property guaranties indistinguishability of those keys and corresponding signatures. It is easy to see, that in the above-mentioned situation, if keys would be from different classes, the adversary would always be able to distinguish between them. 13

14 Since we require a secure signature scheme with flexible public keys to achieve both class hiding and unforgeability under flexible public keys and any signature scheme with re-randomizable keys can achieve at most one of these properties the primitives are clearly seperable. 4 Applications In this section we present natural applications of signatures with flexible public key. First we show how to implement cryptocurrency stealth addresses from signatures with flexible public key which have the additional key recovery property. Then follow generic constructions of group and ring signature schemes. As we will see in Section 5, each of the schemes presented in this section can be instantiated with a signature scheme with flexible public key such that the result improves on the respective state-of-the-art in terms of concrete efficiency, necessary assumptions or both. 4.1 Cryptocurrency Stealth Addresses A direct application of signatures with flexible public keys in the cryptocurrency domain is the implementation of stealth addresses [30]. In cryptocurrency systems such as Bitcoin, transactions are digitally signed, such that the original owner of the funds signs the transaction to transfer funds to another party. In this transaction, the receiving party is also identified by its public key. Using stealth addresses, it is possible for the sender to create a fresh public key for the receiving party from their known public key such that these two keys cannot be linked. The receiving party can recognize the fresh key as its own and generate a corresponding private key, which subsequently enables it to spend any fund send to the fresh unlinkable key. Crucially, there is no interaction necessary between sender and receiver to establish the fresh key and only the legitimate receiver can recover the right secret key corresponding to the fresh key. This can be implemented via a straightforward augmentation of signatures with flexible public keys by a signing key recovery algorithm which allows the holder of the signing key to recover an equivalent signing key from the trapdoor and the fresh public key alone. Scheme 4 is an instances of signatures with flexible public key which achieve this property. We also show how to extend schemes 5 and 6 to support it. 4.2 Group Signatures/Self-blindable Certificates We now present an efficient generic construction of static group signatures that uses SFPK as a building block and which is secure in the model by Bellare, Micciancio and Warinschi [4]. The idea is to generate a SFPK secret/public key pair and certify the public part with a SPS-EQ signature. To sign a message, the signer changes the representation of its SFPK key, and changes the representation of the SPS-EQ certificate. The resulting signature is the SFPK signature, the randomized public key and the SPS-EQ certificate. 14

15 KeyGen GS (1 λ, n) 1 : BG $ BGGen SPS (1 λ ); (pk SPS, sk SPS ) $ KGen SPS (BG, l) 2 : ρ $ CRSGen(1 λ ) / optional 3 : foreach user i [n] : 4 : (pk i, sk i, τ i ) $ TKeyGen(1 λ, ω) 5 : σ i SPS $ Sign SPS (pk i, sk SPS ) 6 : return (gpk := (BG, pk SPS, ρ), gmsk := ([(τ i, pk i )] n i=1 ), 7 : gski := (pk i, sk i, σ i SPS )) Sign GS (gski, m) 1 : parse gski = (pk, sk, σ SPS ) 2 : r $ Z p ; pk ChgPK(pk, r); sk ChgSK(sk, r) 3 : (pk, σ SPS ) ChgRep SPS (pk, σ SPS, r, pk SPS ) 4 : M := m σ SPS pk 5 : σ $ Sign(sk, M) 6 : return σ GS := (pk, σ, σ SPS ) Scheme 2: Generic Group Signature Scheme Opening of signatures work as follows. The group manager generates the SFPK keys with a trapdoor (using TKeyGen) and keeps it along in a list. Note that this means that the group manager s secret key depends linearly on the size of the group. In order to open a signature the manager uses the stored trapdoor to run the ChkRep algorithm thereby determining the equivalence class of the group signature s public key. The group manager can also generate the common reference string ρ $ CRSGen for the SFPK signatures and use it as part of the group public key. This allows us to use schemes which are secure in the multi-user setting, e.g. Scheme 5. Due to space limitations, we only present the setup and signing algorithm for Scheme 2. Verification and opening procedures should be clear from the context. Remark 1 (Self-blindable Certificates). If we use the KeyGen algorithm instead of TKeyGen to compute the SFPK key pair, then there exists no efficient opening procedure and the combination of SFPK and SPS-EQ signature scheme yields a self-blindable certificate scheme [31]. Theorem 3. Scheme 2 is a correct static group signature scheme. Proof. Let λ, n N and let the output of KeyGen GS (1 λ, n) be (gpk = (BG, pk SPS, ρ), gmsk = ([(τ i, pk i )] n i=1), gski = (pk i, sk i, σ i SPS)) Let i [n] and m a message, then Sign GS (gski, m) will output (pk, σ, σ SPS ) where pk ChgPK(pk, r), (pk, σ SPS ) ChgRep SPS(pk, σ SPS, r, pk SPS ) and σ $ Sign(sk, M). Since the relation R F lex is the same for SFPK and the SPS-EQ scheme, ChgRep SPS and ChgPK will output the same pk and because of the correctness of SPS-EQ, Verify SPS (pk, σ SPS, pk SPS) will succeed. Similarly Verify(pk, m, σ) will succeed because of the correctness of the SFPK scheme. Hence verification will succeed. Since the signature was honestly generated, we have pk [pk i ] R, which will be detected by the group manager in the opening procedure by trying all possible trapdoors in the group manager secret key. 15

16 Theorem 4. Scheme 2 is fully traceable if the SPS-EQ signature scheme is existential unforgeable under chosen-message attacks and the SFPK scheme is existential unforgeable. Proof (Theorem 4). We will use the game base approach. Let us denote by S i the event that the adversary wins the full traceability experiment in GAME i. Let (m, σgs = (pk, σ, σsps )) be the forgery outputted by the adversary. GAME 0 : The original experiment. GAME 1 : We abort in case Open GS (gmsk, m, σgs ) = but Verify GS(gpk, m, σgs ) = 1. Informally, we exclude the case that the adversary creates a new user from outside the group, i.e. a new SPS-EQ signature. We will show that this only decreases the adversary s advantage by a negligible fraction. In particular, we will show that any adversary A returns a forgery for which we abort, can be used to break the existential unforgeability of the SPS-EQ signature scheme. The reduction algorithm uses the signing oracle to compute all signature σsps i of honest users. Finally, if the adversary returns (m, σgs = (pk, σ, σsps )), the reduction algorithm returns (pk, σsps ) as a valid forgery. We note that by correctness of the SFPK scheme, if pk is in a relation to a public key of an honest user, then we can always open this signature. It follows that pk is from a different equivalence class and the values returned by the reduction algorithm are a valid forgery against the SPS-EQ signature scheme. It follows that Pr[S 1 ] Pr[S 0 ] Adv l,euf-cma SPS-EQ,A (λ). GAME 2 : We choose a random user identifier j $ [n] and abort in case Open GS (gmsk, m, σgs ) j It is easy to see that Pr[S 1 ] = n Pr[S 2 ]. We now show that any adversary A that has non-negligible advantage in winning full-traceability experiment in GAME 2 can be used by a reduction algorithm R to break the existential unforgeability of the SFPK scheme. R computes all the public keys of group members according to protocol, except for user j. For this user, the algorithm sets pk j to the public key given to R by the challenger in the unforgeability experiment of the SFPK scheme. It is worth noting, that the adversary A is given the group manager s secret key gmsk = ([(τ i, pk i )] n i=1 ). Fortunately, the reduction R is also given τ j by the challenger and can compute a valid secret key gmsk that it gives as input to A. To simulate signing queries for the j-th user, R uses its own signing oracle. By the change made in GAME 2, A will never ask for the secret key of the j-th user, for which R is unable to answer (unlike for the other users). Finally, A outputs a valid group signature (m, σgs = (pk, σ, σsps )) and the reduction algorithm outputs (m σsps pk, σ ) as a valid SFPK forgery. By 16

17 the changes made in the previous games we know that pk and pk j must be in a relation. Moreover, the message m could not be used by A in any signing query made to R. Thus we know that (m σ SPS pk ) was never queried by R to its signing oracle, which show that R returns a valid forgery against the unforgeability of the SFPK scheme. Finally, we have Pr[S 0 ] n Adv euf cma A,SFPK (λ) + Advl,euf-cma SPS-EQ,A (λ). Theorem 5. Scheme 2 is fully anonymous if the SPS-EQ signature scheme perfectly adapts signatures and is existential unforgeable under chosen-message attacks, the SFPK scheme is class-hiding and strongly existential unforgeable. Proof (Theorem 5). We will use the game-based approach. Let us denote by S i the event that the adversary wins the full anonymity experiment in GAME i. GAME 0 : The original experiment. GAME 1 : In this game we change the way we compute the challenge signature σgs Sign $ GS (gsk[i b ], m ). Let σgs = (pk, σ, σ SPS ). We compute (pk, σ) as in the original experiment but instead of randomizing the SPS-EQ signature σ SPS, we compute σ SPS Sign SPS (pk, sk SPS ). Because the SPS-EQ signature scheme perfectly adapts signatures, we have Pr[S 1 ] = Pr[S 0 ]. GAME 2 : We pick a random user identifier j $ [n] and abort in case j i b. It is easy to see that Pr[S 1 ] = n Pr[S 2 ]. GAME 3 : We now abort in case the adversary queries a valid signature (m, σ GS = (pk, σ, σ SPS )) to the Open GS oracle and it fails to open, i.e. the opening algorithm returns. By perfect correctness of the SFPK scheme, it follows that the only way an adversary can make the experiment abort if he is able to create a new user, i.e. create a valid SPS-EQ signature under a public key pk that is not in relation with any of the honest public keys. It follows that we can use such an adversary to break the existential unforgeability of the SPS-EQ signature scheme, i.e. we just use the signing oracle to generate all σsps i and return (pk, σ SPS ) as a valid SPS-EQ forgery. It follows that Pr[S 3 ] Pr[S 2 ] Adv l,euf-cma A,SPS-EQ (λ). 17

18 GAME 4 : We now change the way, we compute the secret key for user j. Instead of using (pk j, sk j, τ j ) TKeyGen FW (λ, ω), we use (pk j, sk j ) KeyGen(λ, ω). Obviously, in such a case we cannot answer the Open GS queries for user j, as the value τ j is unknown. However, we note that if the adversary s query (m, σ GS ) is a valid group signature, then the Open GS must return a valid user identifier (because of the change in GAME 3, we do not return in such a case). Therefore, if there exists no identifier i [n]/{j} for which ChkRep(τ i, pk i, pk ) = 1, we return j. It is easy to note that this is just a conceptual change (because of the change in GAME 3 ) and we have Pr[S 4 ] = Pr[S 3 ]. GAME 5 : We now compute a random SFPK key pair (pk, sk) KeyGen(λ, ω), choose a random blinding factor r, compute public key pk ChgPK(pk, r), secret key sk ChgSK(sk, r) and change the way we compute the challenged signature σ GS = (pk, σ, σ SPS ) under message m. We set M = m σ SPS pk and run σ Sign(sk, M). In other words, instead of using the secret of user i b to generate the signature σ, we use a fresh key pair for this (i.e. a user from outside the system). We note that any adversary that is able to distinguish between GAME 4 and GAME 5, can be used to break the class-hiding property of the SFPK signature scheme. The reduction algorithm can just set one of the public keys from the class-hiding challenge to be part of the public key of the j-th user. In case, the signature given by the challenger in the class-hiding game was created by this user, we are in GAME 4. If it was created by the second user, then we are in GAME 5. Of course, it might happen that the one of the users in the other group member (other than the j-th user) has a public key from the same relation as the second user in the class-hiding experiment. However, this event occurs with negligible probability and we omit it. Lastly, we notice that the challenger in the class-hiding experiment is given the random coins used to generate the secret key to the adversary. Thus, our reduction can reuse those coins and compute the secret key, which he can give to the distinguishing algorithm, as required to fully simulate the anonymity experiment. It follows that Pr[S 5 ] Pr[S 4 ] Adv c-h A,SFPK(λ). The above changes ensure that the challenged signature is independent from the user i b, i.e. we use a random SFPK public key and a freshly generated SPS-EQ signature on it. However, an adversary A can still use the way we implemented the Open GS in GAME 4. Note that in case, he is somehow able to randomize the signature σ GS = (pk, σ, σ SPS ) and ask the Open GS oracle, then we will return i b as the answer. 18

19 We will now show that the adversary cannot create a valid and distinct signature from σ GS = (pk, σ, σ SPS ). Let (m, σgs = (pk, σ, σsps )) be the query made by the adversary and σgs is a randomized version of σ GS. The first observation is that by the change made in GAME 5, we must have that pk and pk are in a relation, otherwise the above attack does not work. Thus, we can use such an adversary to break the strong existential unforgeability of the SFPK signature scheme. Note that by the change made in GAME 5, pk is a fresh public key and the reduction algorithm can use the one from the strong existential uforgeability game. Moreover, in order to generate σ, the reduction algorithm uses its signing oracle. Finally, the reduction algorithm returns ((m σsps pk ), σ ) as a valid forgery. It is easy to see that in case pk pk or σ SPS σsps, the reduction algorithm wins the strong existential unforgeability game. Thus, the only part of the group signature that the adversary could potentially change is σ. This is the SFPK signature and would mean that the adversary was able to create a new signature under the message asked by the reduction algorithm to the signing algorithm. However, the case that σ σ also means that the reduction algorithm breaks the strong existential unforgeability of the SFPK scheme. We conclude, Pr[S 5 ] = Adv seuf cma A,SFPK (λ). Finally, we have ( ) Pr[S 0 ] n Adv l,euf-cma A,SPS-EQ (λ) + Advc-h A,SFPK(λ) + Adv seuf cma A,SFPK (λ). 4.3 Ring Signatures In ring signatures there is no trusted entity such as a group manager and groups are chosen ad hoc by the signers themselves. Thus, we cannot use SPS-EQ to prove that a given ring signature was created by a valid member. Instead we give a membership proof, which is perfectly sound even if the common reference string is generated by the signer. In other words, the actual ring signature is a SFPK signature (pk, σ) and a proof Π that there exists a public key pk Ring that is in relation to the public key pk, i.e. the signer proves knowledge of the random coins used to get pk. The signature s anonymity relies on the classhiding property of SFPK. Unfortunately, in the proof, the reduction does not know a valid witness for proof Π, since it does not choose the random coins for the challenged signature. Thus, we introduce a trapdoor witness. We extend the signer s public keys by a tuple of three group elements (A, B, C) and prove an OR statement which allows the reduction to compute a valid proof Π if (A, B, C) is a non-ddh tuple. More details are given in Scheme 3. We can instantiate this scheme with a membership proof based on the O( n) size ring signatures by Chandran, Groth, Sahai [11] and the perfectly sound proof system for NP languages by Groth, Ostrovsky, Sahai [21]. The resulting membership proof is perfectly sound and of sub-linear size in the size of the set. It follows, that our ring signature construction yields the first sub-linear 19